75-Basics of Trigonometry - 03 Important key points

Click here for the previous basics of trigonometry.

Now we will study the next part of trigonometry. 

Today we will study Angles in standard position, positive and negative angles, the definition of trigonometric ratios in the Cartesian coordinate system, and Signs of trigonometric ratios in different quadrants.

Angles in standard position:

1) Ray OA is an Initial Ray.

2) Rotate ray OA in an Anti-Clock-Wise direction through a certain angle keeping point O as a fixed point.

3) The final position of ray OA is the ray OB which is known as a Terminal ray.

4) Here an angle AOB made by these two rays (Ray OA & Ray OB) with fixpoint O as the vertex is known as the directed angle.

Note: 

1) The rotation of the initial ray OA to the terminal ray OB in an anti-clockwise direction from the positive angle.

2) The rotation of the initial ray OA to the terminal ray OB in the clockwise direction from the negative angle.

Standard angle:

In the Cartesian coordinate system, the amount of rotation of the initial ray OA to the terminal ray OB through an origin is known as the standard angle.

Trigonometric Ratios in the Cartesian coordinate system:

 
1) Let p (x, y) be any point on the terminal ray OP (also called "arm").

2) Let OP = r.

3) By the theorem of Pythagoras, we have 

O A ² + A P ² = O P ² 

 so, x ² + y ² = r ² 

Now we will define all the trigonometric ratios using x, y, and r as shown below.

1) sin q = y / r 

2) cos q = x / r 

3) tan q = y / x 

4) csc q = r / y 

5) sec q = r / x 

6) cot q = x / y 

Signs of trigonometric ratios in different quadrants:                

1) All the ratios in the 1st quadrant are positive.

2) In the 2nd quadrant, "x" is negative, and "y", and "r" is positive. So only the ratios sin and csc contain "y" and "r" so they are positive and the rest are all negative. 

3) In 3 rd quadrant, both "x" and "y" are negative, and "r" is positive. Here, only tan and cot contain "x" and "y". As both "x" and "y" are negative, their division will be positive, so tan and cot will be positive, and the rest all are negative.

4) In the 4th quadrant, "x" and "r" are positive, and "y" is negative. Here only cos and sec contain "x" and "r", so cos and sec are positive, and the rest all are negative.

In short,

      1st  quadrant                    all ratios are positive

      2nd quadrant               "sin" and "csc" are positive

      3rd  quadrant               "tan" and "cot" are positive

      4th  quadrant               "cos" and "sec" are positive 

To remember this case we can prepare the following statements:

1) All Silver Tea Cups                                 (All, sin, tan, cos)     

2) Add Sugar To Coffee                              (All, sin, tan, cos)      

3) All Scholars Take Computer-Science   (All, sin, tan, cos)     

All the students need to remember the above statements so that we will definitely remember the signs of the trigonometric ratios in different quadrants.

The next part of this topic will be published in the next Blog.

Click here for the next basics of trigonometry.

Anil Satpute

74-Basics of Trigonometry - 02 Important key points

Click here for the previous basics of trigonometry.

Trigonometry is an important topic for your higher studies. We must study this topic with all basics. Actually, this topic is very simple. If we concentrate to learn this topic from the beginning, we will definitely understand the basics of this topic and it will help us to remember the formulas (formulae) very easily. At the end of all the Blogs related to “Basics of trigonometry”, we will list all the formulas (formulae) in a separate sheet so that it will be helpful for you to remember all these formulas (formulae).

1) Define all trigonometric ratios for an angle C of the above triangle ABC.  

Answer: 
1) sin C  = AB/AC
2) cos C = BC/AC
3) tan C  = AB/BC
4) csc C = AC/AB
5) sec C  = AC/BC
6) cot C  = BC/AB  

2) Observe the adjacent figure and write down all the trigonometric ratios.
Answer:
1) sin P   = RQ/PQ          1) sin Q   = PR/PQ
2) cos P  = PR/PQ           2) cos Q  = RQ/PQ 
3) tan P   = RQ/PR           3) tan Q  = PR/RQ
4) csc P  = PQ/RQ           4) csc Q  = PQ/PR
5) sec P  = PQ/PR            5) sec Q = PQ/RQ
6) cot P   = PR/RQ           6) cot Q  = RQ/PR  
In the above example, just observe 
case 1) sin P and cos Q
case 2) cos P and sin Q

Here, the basic point to understand is that the comparison of two angles is given with different trigonometric ratios. First, observe these two angles. angle P and angle Q. In a right-angled triangle, QPR, Angle R = 90, so the remaining two angles P and Q are complementary angles. 

Here all of us will remember the following trigonometric relations of complementary angles of a right-angled triangle.

1) sin A  = cos (90-A)
2) cos A = sin (90-A)
3) sec A = csc (90-A)
4) csc A = sec (90-A)
5) tan A  = cot (90-A)
6) cot A  = tan (90-A)

A) An easy way to remember the above relations:

The relation between the trigonometric ratios of co-terminal angles. (See the following figure)

Trigonometric Ratios of an angle A	Trigonometric Ratios of Complimentary Angle of A. (90°-A)	Ratio (In the form of the length of the sides of a triangle)
sin A	cos (90°-A)	BC/AC
cos A	sin (90°-A)	AB/AC
csc A	sec (90°-A)	AC/BC
sec A	csc (90°-A)	AC/AB
tan A	cot (90°-A)	BC/AB
cot A	tan (90°-A)	AB/BC

B) Relation between the trigonometric ratios:

Observe the following trigonometric ratios from the figure given below:

1) sin C  = AB/AC

2) cos C = BC/AC

3) tan C  = AB/BC

4) csc C = AC/AB

5) sec C  = AC/BC

6) cot C  = BC/AB

If we divide equation 1 by equation 2 we get equation 3. i.e. 

                            

In the same way, we can prove that "cos C / sin C  =  cot C"

In the Next Blog, we will study the trigonometric ratios of some standard angles and trigonometric identities.

Click here for the next basics of trigonometry.

Anil Satpute

73-Basics of Trigonometry - 01 Important key points

When an artist wants to draw a picture then he/she always draws a replica of that object. Say, a replica of the great Taj Mahal of Agra (India) or the Statue of Liberty and so on. He/She fixes the drawing paper on the table. Fixes the distance between the paper and his/her eyesight. Then he/she takes all required measurements of the object (Taj-Mahal/Statue of Liberty) on a pencil by holding it between his/her eye-site and the object by locating the bottom of an object at the lower side of the pencil and the tip of an object at the tip of the pencil. Using these distances on the pencil he/she fixes the height, width, and all other dimensions of an object on the drawing paper. Here, using the properties of similarities in a right-angled triangle, an artist or an architecture

can able to draw the exact replica of an object in a smaller size. He/she holds the tip of the pencil adjusting with the tip of the object and fixes the bottom of the object at the lower part of the pencil to fix the height of an object. In the adjacent figure, an artist/architect observes an object BC from point A and marks its height as MN on the pencil. Here ABC is the right-angled triangle with AB as the base, BC as the height, and AC as the hypotenuses. Here the line segments BC and MN are in proportion with AB and AM.
Mathematically, we can say that [BC/AB] = [MN/AM].

Now let us study something about the right-angled triangle. In the adjacent figure triangle, ABC is the right-angled triangle. < ABC is right-angled, so the side opposite to the right angle is known as the hypotenuse. Considering < A is our angle of the right-angled triangle, then side AB is known as the side adjacent to angle A, and side BC is the side opposite to angle A.

Note: There are two types of concepts of trigonometry. We will consider the first concept as a lower-level concept of trigonometry and the other one as a higher-level trigonometry. In lower-level trigonometry, we will study the trigonometric ratios of only the acute angles of the right-angled triangle. Whereas in higher level trigonometry, we will study the trigonometric ratios for any angle. Now let us start with lower-level trigonometry. 

Now, let us start our study of trigonometry. For any right-angled triangle, there are three sides. Let us write down all possible ratios of the lengths of the triangle. 

1) BC/AC

2) AB/AC

3) BC/AB

4) AC/BC

5) AC/AB

6) AB/BC

There are only 6 possible ratios that can be defined from triangle ABC. Considering these 6 ratios, we can develop entire trigonometry. Trigonometric ratios can be defined for a certain angle of the right-angled triangle. Let us understand the words "Side opposite to an angle" or "Side adjacent to an angle". See the above diagram carefully. In the above diagram, we consider angle A as our angle of the right-angled triangle ABC. Side AB is adjacent to angle A and in the same wayside BC is the side opposite to angle A. We will write the above six ratios considering angle A once again using the words "Side opposite to an angle", "Side adjacent to an angle" and "hypotenuse". We will name these ratios in trigonometric forms.

1) BC/AC   side opposite to A/hypotenuse (sin A)

2) AB/AC   side adjacent to A/hypotenuse (cos A)

3) BC/AB   side opposite to A/side adjacent to A      (tan A)

4) AC/BC   hypotenuse /  side opposite to A                 (csc A) 

5) AC/AB   hypotenuse /  side adjacent to A                 (sec A) 

6) AB/BC   side adjacent to A/side opposite to A      (cot A)

Details of trigonometric ratios:
1) sin A  ------  sine of angle A
2) cos A ------  co-sine of angle A
3) tan A  ------  tangent of angle A
4) csc A ------  co-secant of angle A (generally written as cosec A)
5) sec A ------  secant of angle A
6) cot A ------  co-tangent of angle A

Now we will define all six trigonometric ratios as follows:

1) sin A   =  BC/AC  =  opposite / hypotenuse
2) cos A  =  AB/AC  =  adjacent / hypotenuse
3) tan A   =  BC/AB  =  opposite / adjacent
4) csc A  =  AC/BC  =  hypotenuse / opposite
5) sec A  =  AC/AB  =  hypotenuse / adjacent
6) cot A  =  AB/BC  =  adjacent /  opposite

Now study all basic concepts of trigonometry as discussed above and do at least two experiments as stated below.

1) Define all trigonometric ratios for an angle C of the above triangle ABC.
2) Observe the adjacent figure and write down all the trigonometric ratios.

Click here for the next basics of trigonometry.

Anil Satpute

72-Magic Square-5 (different cells and addition part-2)

Click here for the next blog on the magic square.

Addition of different cell elements of Magic Squares:

🔢 Delving Deeper into Hidden Patterns in the Magic Square  

We have already observed that in a 4 × 4 magic square, the total numbers in every row, column, and diagonal consistently add up to 34.  

That’s the charm of a magic square!  

Each of these lines comprises four elements; this unchanging total is one of its key characteristics.  

However, what’s even more intriguing is that the sum of numbers arranged in particular patterns or positions throughout the square, in addition to rows, columns, or diagonals, also totals 34.  

📐 The illustration below highlights various combinations of 4 cells that adhere to this unique property.  

It’s akin to uncovering concealed symmetry and mathematical harmony that goes beyond the obvious.

 (1 + 15 + 12 + 6)      (14 + 4 + 7 + 9)

(R₁C₁, R₁C₂, R₂C_1, R₂C₂)    (R₁C₃, R₁C₄, R₂C_3, R₂C₄)

🔍 Discovering More Concealed Groups in the Magic Square

In the provided example, the total of the chosen numbers — 1 + 15 + 12 + 6 — equals 34.

This verifies the magical characteristic even when numbers are selected from non-linear positions!

Similarly, numerous other combinations of 4 cells, located in various square areas, also yield the same sum.

✨ Now let's delve into additional groups where the numbers may not align in a straight line, yet their sum still fantastically amounts to 34.

These instances beautifully showcase the hidden symmetry and refined structure of a magic square.

 

(11 + 5 + 2 + 16)      (8 + 10 + 13 + 3)

(R₃C₃, R₃C₄, R₄C_3, R₄C₄)  (R₃C₁, R₃C₂, R₄C_1, R₄C₂)

(15 + 14 + 3 + 2)      (8 + 12 + 9 + 5)

(R₁C₂, R₁C₃, R₄C_2, R₄C₃)  (R₂C₁, R₃C₁, R₂C_4, R₃C₄)

(15 + 12 + 5 + 2)      (14 + 9 + 8 + 3)

(R₁C₂, R₂C₁, R₃C_4, R₄C₃)  (R₁C₃, R₂C₄, R₃C_1, R₄C₂)

(6 + 7 + 10 + 11)         (1 + 4 + 13 + 16)

(R₂C₂, R₂C₃, R₃C_2, R₃C₃)  (R₁C₁, R₁C₄, R₄C_1, R₄C₄)

(1 + 7 + 10 + 16)         (4 + 6 + 11 + 13)

(R₁C₁, R₂C₃, R₃C_2, R₄C₄)  (R₁C₄, R₂C₂, R₃C_3, R₄C₁)

(6 + 9 + 3 + 16)         (15 + 4 + 10 + 5)

(R₂C₂, R₂C₄, R₄C_2, R₄C₄)  (R₁C₂, R₁C₄, R₃C_2, R₃C₃)

(12 + 7 + 13 + 2)         (1 + 14 + 8 + 11)

(R₂C₁, R₂C₃, R₄C_1, R₄C₃)  (R₁C₁, R₁C₃, R₃C_1, R₃C₃)
✨ Explore More Magical Patterns!
In the same way, you can discover many intriguing relationships between the numbers placed in different cells and their resulting sums. Keep exploring — every pattern reveals a new layer of mathematical beauty!

Click here for the next blog on the magic square.

ANIL SATPUTE

71-Magic Square-6 (Jumbo magic square)

Click here for the next blog on the magic square.

A) Magic square using sketches of 3x3 on 4x4

🧮 Delving into the Realm of Jumbo Magic Squares  

We've previously examined how to create a magic square using a magic sketch, an innovative and visual approach that introduces order and clarity to intricate configurations.  

Now, let’s elevate our exploration!  

By combining different types of magic sketches, we can construct magic squares of significantly larger dimensions. These impressive creations demonstrate numerical balance and showcase the captivating aesthetics of mathematical design on a grander scale.  

These are Jumbo Magic Squares, a harmonious combination of structure, symmetry, and magnitude.  

🔷 Whether a 16×16 square or even larger, the Magic Sketch method enables us to construct them systematically and creatively.  

🔷 Each digit is strategically positioned to create rows, columns, and diagonals that yield the same magical sum.  

Therefore, if you enjoy numbers and patterns, crafting Jumbo Magic Squares offers a true pleasure, both a mental challenge and a visual enjoyment!

🧠 Creating a 12 × 12 Jumbo Magic Square with 3 × 3 and 4 × 4 Magic Sketches  

In the captivating realm of Magic Squares, possibilities are endless. We can construct larger and more complex magic configurations by utilizing smaller, manageable units, such as a 12 × 12 jumbo magic square.

Let’s explore how this can be achieved by combining two familiar magic sketches: 3×3 and 4×4.

🔷 Two Effective Methods to Construct a 12 × 12 Magic Square:  

1️⃣ Method One: Starting with a 4×4 Magic Sketch  

Initiate with the 4 × 4 Magic Sketch; this will serve as your foundational layout.

Now, envision each point in the 4 × 4 base sketch as the beginning point for a 3 × 3 square of houses.

This means each point of the 4 × 4 sketch transforms into a 3 × 3 magic square, resulting in:

4  x 3 = 12

(for both rows and columns)  

4×3=12 (both rows and columns)  

Therefore, the overall square expands to a 12 × 12 grid of 144 magic numbers!

🔗 Arranging the Squares:  

Commence from point “A” in the first 3 × 3 sketch.

Link it to point “B” of the subsequent 3 × 3 sketch, following the pathway from your 4 × 4 base.

Proceed with this meticulous connection for all 16 positions in the 4 × 4 sketch.

This visual linking culminates in a Magic Sketch for 12 × 12, preserving structure and numerical balance.

📐 Refer to the diagram below for a clear depiction of how these connections evolve to finalize the Jumbo Magic Square.

In this section, we can create two varieties of magic squares.  

1) Gradually build up the magic square by incorporating a 3x3 pattern onto a 4x4 template.  

Here, the foundational sketch is a 4x4 magic sketch, and we are using this base to apply the 3x3 magic design. 

2) A consistently ascending magic square with a 3x3 design on a 4x4 template. This foundational sketch is a 4x4 magic template, and we are integrating the 3x3 magic design onto this base.

B) Magic square using sketches of 4x4 on 3x3

🔷 Method 2: Utilizing a 3 × 3 Magic Sketch as the Foundation for Creating a 12 × 12 Jumbo Magic Square  

Similar to piecing together a stunning mosaic, we can develop a Jumbo Magic Square (12 × 12) using a different method, starting with a 3 × 3 Magic Sketch as our base.

🧩 Step-by-Step Assembly:  

Start with the 3 × 3 magic sketch, our initial layout.

Next, each point in this 3 × 3 framework will be expanded into a 4 × 4 Magic Square containing 16 houses.

Given that:

3 x 4 = 12

(for both rows and columns)

3×4=12 (for both rows and columns)

We will construct a complete 12 × 12 magic square that contains 144 numbers.

🔗 Linking the Sketch:  

Assuming in your foundational 3 × 3 sketch, you have the first point labeled “A.”

Utilize your 4 × 4 magic sketch to determine the movement:  

From point “B” in your magic sketch, trace a path that connects to the “A” of the subsequent point in the base sketch.

Repeat this connection pattern for all 9 positions of the 3 × 3 foundation.

Doing so ensures that each 4 × 4 block aligns correctly with the design and flow of the original 3 × 3 sketch, resulting in a Magic Sketch for the entire 12 × 12 grid.

📐 A well-drawn diagram will help you visualize how each 4 × 4 square integrates into the larger sketch, ultimately leading to the completion of a Jumbo Magic Square!

🔶 Building a 12 × 12 Jumbo Magic Square Using the “4 × 4” Base Sketch

In our previous method, we used the 4 × 4 Magic Sketch as the base and extended each point to form a 3 × 3 magic square, yielding a beautifully crafted 12 × 12 Jumbo Magic Square.

Now, let’s continue the explanation with more detail.

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🔁 Completing the Full Sketch:

• Start with the 4 × 4 Magic Sketch (which contains 16 points).

• Each of these 16 points will be replaced by a 3 × 3 square, each containing 9 houses.

• These small 3 × 3 blocks will be carefully filled with numbers so that, when combined, they form a fully functional 12 × 12 magic square, since 4 × 3 = 12.

👉 Process of Expansion:

• Consider a point “A” in the base 4 × 4 sketch.

• Attach the “B” point (from your smaller 3 × 3 sketch) to the next “A” of the next square; this defines the direction of progression.

• Repeat this connection and placement until all 16 squares are aligned, reaching the last point of the base sketch.

---

🌟 What Do We Get?

Once complete, we achieve a jumbo magic square made up of 144 cells (12 rows × 12 columns) that maintains the magic properties of the same row, column, and diagonal sums thanks to the underlying logic of magic sketches.

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🧠 Two Unique Possibilities:

Using this idea, we can prepare two distinct types of magic squares:

1. Type 1 – 4 × 4 base with each point as a 3 × 3 Magic Square
   Result: 12 × 12 Jumbo Magic Square

   • Focuses on the 4 × 4 movement.

   • Builds complexity on finer 3 × 3 arrangements.

2. Type 2 – 3 × 3 base with each point as a 4 × 4 Magic Square
   Result: Another version of 12 × 12 Jumbo Magic Square

   • Compact path movement of the 3 × 3 base.

   • Detailed 4 × 4 filling at each node.

Each method is equally elegant and exciting to explore, revealing the limitless creative potential of numbers and mathematical patterns.

1) Gradually enhance the magic square by incorporating a 4x4 design onto a 3x3 layout. The foundational sketch is a 3x3 magic design, and we are building a 4x4 magic design on top of it.

2) Uniformly escalating magic square featuring a 4x4 design on a 3x3 template. The foundation is a 3x3 magic layout, and we are building on this framework to incorporate the 4x4 magic design.

Click here for the next blog on the magic square.

ANIL SATPUTE